J . A m . Chem. SOC.1985, 107, 6851-6859 the amino acid sequence, a rather detailed characterization of the metal cluster architecture was obtained. In particular, it is worth noting that two of the Cd2+ ions of the four-metal cluster, i.e., Cdl and Cd7, are bound to cysteines in relative sequence positions i,i 2 and i 3. In further work on the structural interpretation of the present experimental data (Figure l), the conformational constraints implied by the 'I3Cd-'H and 113Cd-"3Cd scalar couplings are being used as input for the determination of the spatial structure of MT-2 with distance geometry calculationsls in combination with 'H-'H NOE distance constraints and the complete, sequence-specific 'H N M R assignments for the polypeptide chain. Comparison of the presently obtained results (Figure 7) with the available literature shows on the one hand that the assignment of the Il3Cd2+resonances to the two clusters and the 113Cd-1'3Cd connectivities within the clusters are consistent with the previous proposal made by Otvos and Armitage on the basis of 1D homonuclear Il3Cd decoupling experiment^.^,^ On the other hand, in a hypothetical model for MT-2 proposed by the same group17 which included sequence-specific identification of the cysteines bound to the individual Cd2+ions, 18 of the proposed 28 Cd-tocysteine connectivities are incompatible with the present experiments, including all the proposed bridging cysteines. After this manuscript was completed, we became aware that Otvos et a1.28and Live et aLZ9applied 2D N M R experiments
+
+
(28) Otvos, J. D.; Engeseth, H. R.; Wehrli, S. J . Magn. Reson. 1985, 61, 579-584. (29) Live, D.; Armitage, I. M.; Dalgarno, D. C.; Cowburn, D. J . Am. Chem. SOC.1985, 107, 1775-1777.
685 1
related to those in Figure 2 for studies of I13Cd-lH J connectivities in MT-2 and crab metallothionein, respectively. Their experimental schemes differ from those of Figure 3 by the absence of purging procedures. There are no obvious discrepancies between the II3Cd-IH couplings identified by Otvos et aLZ8and in the present paper. The comparison of the data is limited, however, since in those studiesZ8the experiments were recorded at lower frequencies and the spectra were presented in the absolute value mode with reduced spectral resolution, and no resonance assignments other than the seven Cd were gi-~en.~' Acknowledgments. We thank M. Sutter for the preparation of the biological material and E. H. Hunziker and R. Marani for the careful preparation of the figures and the manuscript. This research was supported by the Kommission zur Forderung der wissenschaftlichen Forschung (Project 1 120), the Schweizerischer Nationalfonds (Projects 3.284-82, 3.207-82, and 2.441-82), and the Science and Engineering Research Council (UK) (Overseas postdoctoral fellowship to D. N.). Registry No. 'I3Cd, 14336-66-4;Cd, 7440-43-9;Cys, 52-90-4. (30) Note added in proof After this paper was submitted, we were informed that a homologous metallothionein had been studied by crystallographic methods. The crystal structure of rat liver metallothionein isoform 2 has a substantially different arrangement of cysteines than reported here. The metallothionein in the crystals contains five Cd and two Zn per mole of protein (Melis, K. A.; Carter, D. C.; Stout, C. D.; Winge, D. R. J . Biol. Chem. 1983, 258, 6255-6257. Atomic coordinates for 414 atoms (61 amino acids, 7 metals) derived from a 2.3-A resolution electron density map were deposited June 12, 1985 with the Protein Data Bank, Brookhaven National Laboratory, Upton, NY 11973 (Furey, W. F.; Robbins, A. H.; Clancy, L. L.; Winge, D. R.; Wang, B. C.; Stout, C. D.; manuscript in preparation).
Superdegenerate Electronic Energy Levels in Extended Structures Timothy Hughbanks Contribution from the Department of Chemistry, The University of Chicago, Chicago, Illinois 60637. Received February 22, 1985
Abstract: Near singularities may occur in the electronic density of states of crystalline compounds under circumstances described in this paper. Such "superdegeneracies"are described as they result from simple Hiickel treatments of various systems. Although these superdegeneraciesare accidental in that they are broken when interactions ignored in the simple Hiickel model are restored, large peaks in the density of states remain. The concepts presented are applied to both real and hypothetical cases. Superdegenerate bands are shown to invariably have a nonbonding character which can be fully understood only by consideration of the orbitals available for bonding in the extended structure. While the bands which cause superdegeneracies are flat, the Wannier functions associated with these bands cannot be well localized. The physical implications of this poor localization are discussed. In cases where superdegeneratebands are half occupied, ferromagnetic ground states appear to be favored. Analogies to molecular cases. where the importance of localizability of Hiickel nonbonding molecular orbitals has already been closely examined, may point the way for the extension of this work beyond the simple one-electron treatment given here.
In chemistry and physics, particular phenomena are described by theoretical models that are meant to serve as prototypes of real systems. The features of the models that are associated with these phenomena are often implicitly assumed to be necessary conditions for the Occurrence of the phenomena in question. Such is the case in the explanation of narrow electronic energy bands and the concomitant peaks associated with these bands in crystalline solids. Narrow bands naturally arise when overlaps between atomic orbitals centered on neighboring atoms throughout a crystal are small. In the regime of small interatomic or intermolecular interaction, crystalline charge densities will differ little from those of the constituent atoms, molecules, or ions. The outgrowth of these observations is the assumption that narrow energy bands 0002-7863/85/ 1507-6851$01.50/0
are to be found only when a system is characterized by at least one set of weakly interacting atomic or molecular orbitals. Simple physical prototypes in solid-state theory often ignore the kinds of subleties that specific structures may possess. This paper presents a discussion of cases in which flat bands are a consequence of the pseudo-symmetry of the extended structure of a solid as a whole. The underlying reasons for the Occurrence of these highly degenerate bands are revealed by examining systems in the light of the simple Hilckel model and its ability to uncover features that have a topological origin. The presentation is meant to be suggestive, and it is hoped that it will stimulate investigation into the ways in which particular extended structures show unusual properties because of these superdegenerate bands. 0 1985 American Chemical Society
6852 J . Am. Chem. SOC., Vol. 107, No. 24, 1985
Hugh banks
A Prototypical Example In conventional LCAO treatments of crystalline solids’ (i.e., tight-binding methods), one begins with a symmetry-adapted basis set of orbitals (@,(k)]that are Bloch sums of AO’s, (4,(r-R)), of the N unit cells of the crystal:
The sum over R extends over the unit cells of the crystal; the function @@(r-R)is the pth A 0 of a unit cell associated with the lattice site specified by R. The factor eik.Ris the phase change in the orbitals {&(k)] on moving from a given reference cell (at the origin) to a cell at the site specified by R. The set {d,(k)) is symmetry adapted in that the translational symmetry has been fully exploited in employing this basis. Thus, the full crystal orbital problem is divided into separate problems for each wavevector k, each problem having the dimension of the number of AO’s per unit cell. The periodicity of eik.Rwith respect to k allows one to restrict k to the first Brillouin zone in k space. Further details are available in standard texts.lb.c Crystal orbitals obtained by solution of the secular equation using the Bloch basis functions of eq 1 are expressed as a linear combination of Bloch functions for each k: +n(k) = Cc,n(k)4,(k)
(2)
P
where n is the so called “band index” and runs from 1 to the number of atomic orbitals in the unit cell, as does p. At this point it should be noted that the set of Bloch basis functions, {&(k)), need not be built from AOs, but may consist of MOs or fragment MO’s or any other convenient (but equivalent) basis. In particular, we may choose to transform to a basis set that is explicitly k dependent. We may, for example, wish to make a change in basis involving 4,(k) and ~ $ ~ ( k ) : dl’(k) =
42’(k) =
1
d -
d-
[‘#‘l(k) + AkdZ(k)l
1
+ 4Z(k)1
produce a qualitative spectrum of energy levels for planar aromatics has a time-honored place in the training of chemists. Therefore, to first demonstrate the phenomenon of superdegeneracy in an extended system, we will consider a hypothetical problem in an extended two-dimensional net. Consider the net, 1, consisting of an infinite collection of fused three- and nine-
1
membered rings. One may think of this as an unlikely alternative to graphite for which we wish to know the T energy levels. The simplest way to proceed in this problem is to divide the unit cell into fragments of which it is composed. Therefore, we will proceed by considering the interaction of the K molecular orbitals of the triangle (of all’ and e” symmetry) with the p~ orbital (called 4) of the single trigonal atom that links these triangles together (see 2). We will refer to the Bloch basis orbitals corresponding to
(3a)
(3b) 2
where X,I is a complex number that varies with k. Naturally, the coefficients in eq 2 are transformed in the new basis as well since +,(k) will not depend on the basis: cl’(k) =
d-
1
1
C
pb== 10
ii
known structure of AZM3S4compounds8 except that in the latter (8) (a) Huster, J.; Bronger, W. J . Solid State Chem. 1974. 1 1 , 254. (b) Giinther, 0.;Bronger, W. J . Less Common Met. 1973,31, 255. See also the cyclic hexamer [Ni(SC2H5)2]6reported in: (c) Woodward, P.; Dahl, L. F.; Abel, E. W.; Crosse, B. C. J . A m . Chem. SOC.1965, 87, 5251.
6856 J . Am. Chem. SOC.,Vol. 107, No. 24, 1985
Pt d
contribution
As
ff.contribution
*I
Figure 4. T h e As, T * and P t in-plane contributions to the density of states for a two-dimensional Pt,(As,)?- layer are shown. The “in-plane” d orbital is defined relative to the adjacent As, dimers as depicted in 8. N o t e the position of the Fermi level and the superdegenerate As2 K* band.
no S-S bonds are found. The As, coordination suggested in this structure has precedent in [M(C0)5]3(X2)molecules (M = W or Mo; X = As, Sb, or Bi); see 11. Such a system may also be A
12
d
viewed as a hexagonal analogue to the tetragonal structure recently reportedlo for BaPd,X, (X = P, As) in which X, dimers are found in layers as illustrated in 12. The Pd centers have the same
00
I’
0
d
11
(9) (a) Sigwarth, B.; Zsolnai, L.; Berke, H.; Huttner, G. J . Organomef. Chem. 1982, 226, C5. (b) Huttner, G.; Weber, U.; Sigwarth, B.; Scheidsteger, 0. Angew. Chem. 1982,94,210; Angew. Chem., In?. Ed. Engl. 1982, 21,215. (c) Huttner, G.; Sigwarth, B.; Scheidsteger, 0.;Zsolnai, L.; Orama, 0. Organomefallics 1985, 4, 326. (10) Mewis, A. Z . Naturforsch., Teil B 1984, 39, 713.
Hughbanks bow-tie coordination found in K,PtAs2 and in the proposed hexagonal structure 10. The t[M3(X2)22-]net contains two X2 dimers and thereforefour P* orbitals per unit cell. By reasoning exactly parallel to that outlined in treating the t [ P t ( A ~ , ) ~ -chains, l we know there are three metal d orbitals which may effectively interact with them. Therefore, one As2 K * band will be left nonbonding and will lie relatively low in energy. An extended Hiickel calculation on this system underscores this conclusion. Figure 4 shows the As, K* and Pt in-plane d contributions to the density of states for a t[Pt3(As2)22-]system. There is considerable mixing of these levels throughcut the energy range except that the peak present just below the Fermi level in the As2 K* contribution has no counterpart in the Pt d part. Just as in the t[Pt(As,)*-] chains, an As2 nonbonding K * band is the valence band.” Because of the detailed nature of the Pt d-As, K * antibonding bands, they extend down to “touch” the nonbonding band and this material should be metallic. The above cases are ones in which the requirements for superdegenerate bands are loosened somewhat with the dividend of some generalization. We can infer the existence of nonbonding bands in an extended system merely by inspection of the extended structure and careful consideration of the orbitals available for bonding. The most ambitious goal is the realization of a system in which the superdegenerate bands are quite narrow and only partially occupied. One hypothetical but plausible candidate may be constructed (on paper) from the 9-3 net by an isolobal replacement.’, The crucial ingredient of the 9-3 net that led to a superdegenerate band is the set of e” orbitals of the threemembered ring. If we replace the three-membered ring with a linear ML2 group, two d orbitals may take on the role of the e” ring orbitals (see 13). The structure that results is shown in 14
v
T
L
w L
13
and is a layer structure held together by linkages which form a honeycomb array.” If this structure were adopted by a compound in which the transition metal has a d’ configuration the results might prove very interesting. In Figure 5 the P DOS is plotted for the case where X = P, M = Mo, L = H. Such a compound would be nearly ideal because the relatively long Mo-P distance (taken here to be 2.3 A) mandates a very weak through-space Mo-Mo interaction (Mo-Mo = 3.98 A). Figure 5 shows an obvious resemblance to Figure 1, only here the superdegenerate band arises from Mo d orbitals. What would be the nature of the electronic ground state? Before addressing this question we must first consider the possibilities for localization of superdegenerate band states. (1 1) Actually, a band which is symmetric with respect to reflection in the plane of the layer overlaps with the nonbonding antisymmetric T* band and makes a definitive assignment of the valence band impossible. Of course, this does not alter the point being made here. (12) Hoffmann, R. Angew. Chem. 1982, 21, 711, and references therein. (13) Such layers are imbedded in couellite CuS; see: na) AlsCn, N. Geol. Foeren. Sfockholm Foerh. 1931, 53, 11 1. (b) Oftedal, I . Z . Kristallogr. Mineral. 1932, A83, 9. (c) Hulliger, F. Sfruct.Bonding (Berlin) 1968, 4, 83. (d) Pearson, W. B. “The Crystal Chemistry and Physics of Metals and Alloys”; Wiley: New York, 1972.
J . Am. Chem. Soc., Vol. 107, No. 24, 1985 6857
Superdegenerate Electronic Energy Levels
0
n
b
14
”1’
t
hloH,P: tolal n DOS
310 dizYz
J’
1h
Figure 5. A t left the total
( n = band index)
(10)
k
-
When $,(k) is overwhelmingly dominated by the contribution made by a single molecular orbital ($,(k) CRe’k‘Ro(r-R)), then the form of the Wannier functions approximately reduces to that o(r-R)). In the case of the of the molecular orbitals (w,(R) superdegenerate bands discussed above, it is not possible to construct such localized Wannier functions even though superdegenerate bands are flat (perfectly so, in the simple Huckel model). We will discuss this feature utilizing the R e o , case as an illustration of the general points to be made. The simplest way to understand the nature of the Wannier functions for the superdegenerate oxide p band in R e o 3 is to recognize that they are constrained to be orthogonal to the Re d, orbitals. It can be seen by inspection that the combination of p orbitals illustrated in 15a is the most localized such com-
-
t
-14
Localization When narrow bands result from the weak interaction between orbitals of the constituent molecules of a crystal, it is correct to assume that the crystal orbitals associated with these bands are little different from their molecular parents. An orthonormal set of functions known as Wannier functionsI4 (defined as in eq 10) are very useful in discussing the transition from delocalized to localized states. They can be made optimally localized by a suitable choice of phases for the crystal orbitals for the band in question. 1 wz w,(R) = -Ce-ik’R$,(k)
c
-8
n
T DOS for a MoH,P layer is shown. I n the center and right panels respectively the Mo [d,,,dY,} and P pr contributions are shown. Note that the Fermi level for a d’ system is situated in the middle of the superdegenerate d band.
metal centers included in the illustration. However, a set of functions such as 15a centered at each lattice sire are not mutually orthogonal; functions centered on neighboring sites have an overlap equal to 1/4. Since the Wannier functions are an orthonormal set, orthogonalization of these functions must be carried out to obtain their correct (and maximally localized) form.15 This orthogonalization is straightforward if somewhat tedious, the obvious result being an even further delocalization of the Wannier functions beyond four centers, as shown in 15b. The delocalized Wannier functions discussed above should be contrasted with the more usual situation that obtains for narrow band systems. Consider, for example, the simplest of extended systems, a chain of H atoms. If the interatomic spacing is sufficiently wide, then we certainly expect the electrons to localize; that is, an electron will prefer to remain on a given center rather than experience the electron repulsion which will result if it shares a site with another electron. Put another way, if the range of energy between the most bonding and antibonding orbitals is small (as it will be when interatomic overlap is small), electron repulsion will cause the electrons to localize. This behavior is reflected in the nature of the Wannier functions for such a system. I f interatomic overlap is small, then neighboring s orbitals can be “mutually orthogonalized” by adding a small contribution to each orbital from the neighboring sites, as illustrated in 16a. When
b
V
-
O
-
-
-
16
a
15
b
bination which is indeed orthogonal to the d, orbitals on the four (14) (a) Wannier, G. H . Phys. Rev. 1937, 52, 191. For a review giving many results on Wannier functions, see: (b) Blount, E. I. “Solid State Physics, Advances in Research and Applications”; Seltz, F., Turnbull, D., Ehrenreich, H., Eds.; Academic Press: New York, 1962; Vol. 13, 305. Also see: (c) des Clolzeaux, J. Phys. Rev. A 1964, 135, 685, 698. (d) Monkhorst, H. J.; Kertesz, M. Phys. Rev. B: Condens. Matter 1981, 24, 3015.
the atoms are brought close together, the interatomic overlap grows as does the width of the s band. The electrons will now be more appropriately described as occupying delocalized orbitals starting with those which are most bonding. The poorer localization of the Wannier functions reflects this change; the tails of these functions must grow longer to ensure their mutual orthogonalization (see 16b). This is in contrast to the characteristics we have described for narrow superdegenerate bands; while these bands are quite narrow, the Wannier functions remain as delocalized as in the wide band case. The above results call into question the often made implicit assumption that Wannier functions associated with narrow bands (15) Further localization is possible if one abandons the requirement of orthogonalization; see: (a) Anderson, P.W. Phys. Rec. Lett. 1968, 2 1 , 13. (b) Anderson, P. W. Phys. Rep. 1984, 110, 31 1.
6858
J . Am. Chem. SOC.,Vol. 107, No. 24, 1985
Hughbanks
are necessarily highly 1ocalized.I6 One area in which this may have an important physical effect is in the formation of excitons by the promotion of an electron from a valence band in a semiconductor to the conduction band. If the valence (or conduction) band is superdegenerate then the hole (or electron) in the excitonic state may be considerably more delocalized than would have been expected solely on the basis of the width of the superdegenerate band involved. Further localization of the hole (or electron) is possible by mixing contributions in from bands not formally involved in the excitation, but this will be limited by the cost of raising the "orbital energy" contribution to any excitons formed. For example, in perovskites such as BaTi03 or K N b 0 3 (related to R e o , as discussed above), excitations from the valence (superdegenerate) 0 p levels may include the formation of excitons, but the localization of the hole will not be as great as might be expected from the narrow width of the valence band. The nature of systems in which superdegenerate levels are partially occupied should be more interesting. Insight into these cases is gained by drawing on analogies to molecular diradicalsl' in which two electrons occupy doubly degenerate (usually nonbonding) molecular orbitals. For example, let us contrast the behavior of cyclobutadiene (CB) and trimethylenemethane (TMM). Energy level schemes for the R electrons in the planar species are presented in 17. The relative energies of the states
D-
17
derived from the ground configurations for both of these species have been the subject of intense experimental and theoretical scrutiny. However, it is instructive to examine the simple state diagrams shown in 18. In CB one expects both the 3A2, and the lA1g
f
'AI
tI
2Yy t t
Jaa-Jab
I
lE'
P
As
Mo
orb. 3s 3P 4s
4P 5s 5P
4d 6s 6P 5d "Exponents: Pt
Hi, (eV)
61
(C,Y
eg orbitals can be completely localized on different centers (as illustrated) while in T M M the e" orbitals must spread over common centers. The triplet wave functions for both molecules are such that the electrons avoid occupying the same AO's at the same time (by the Pauli principle). Only for CB where the e, and eb functions spread over different atoms can a singlet state (IBlg) also avoid on-site repulsions." The systems we have discussed are clearly like TMM: a localized set of orbitals derived from superdegenerate band orbitals have large intersite overlap densities. We can predict that a half-filled superdegenerate band should lead to a ferromagnetic (Le., high-spin) ground electronic state*O in order to avoid the large electron-electron repulsion which will occur when the Wannier functions have large intersite overlap densities. Fundamentally, this is just an extension of ideas put forward for molecules and rests on a simple criterion concerning the localizability of the Hiickel MO's for the system under consideration.
=
Appendix I Some of the analytical details of the R e o 3 problem will now be presented. We take up the discussion which was cut short following eq 8 in the text. The orthonormal combination of p,(k) and p,(k), to be called @(k),will interact with d,(k), and a 2 X 2 secular equation can be set up involving these functions. For X(k) and 4(k), we obtain: x(k) =
JwJxy
- Jxx-Jxy
1
.\/4 - 2
COS
(k,a) - 2
[(l COS
J2-g
3
18
( 1 - e-lk
@(k) =
1
4-2
COS
( k , ~ -) 2
)P,(k)l ( A I )
(1
[(l - &y")p,(k) COS
+
(k,a)
and the secular equation involving d,(k)
and 4(k) is
TMM
IB1, states to lie low in energy, separated only by a small exchange splitting: 2Kub. In TMM, however, the 3A21state is expected to lie well below the lowest singlet, 'E', by a large exchange energy: 2Kxy. As pointed out by Borden and Davidson,Is the difference in these two splittings is due to the fact that in CB the degenerate ~
e-'kx")p,(k) -
(1 - eik~")p,(k)l (A21
t
CB
-
(k,a)
2 K a b --I-3*
.i-2 ( C d Q
-18.6 1.88 -12.5 1.63 -16.22 2.23 -12.16 1.89 -8.77 1.96 -5.60 1.90 -1 1.06 4.54 (0.5899) 1.90 (0.5899) -9.08 2.55 -5.47 2.55 -12.59 6.01 (0.6334) 2.696 (0.5513) double-( d functions are used for transition metals.
Acknowledgment. Thanks are due to the Dow Chemical Co. for their support of this research. I owe special thanks to Professor J. K. Burdett for his support and encouragement and for helpful discussions. Thanks are also due to Stephen Lee for first bringing the superdegeneracy found in the 9-3 net to my attention and stimulating this work.
TM M
cr3
Table I. Parameters for EH Calculations
where t d and tp are respectively the Re(d) and O(p) orbital energies. The wavevector k ranges over the Brillouin zone, and in all of these equations it is sufficient to restrict one's attention to the region of k-space in which - a / a Ik,,k, I* / a . The magnitude of Pdp(k)is related to the Hiickel d r p a resonance integral Pdpby the relation:
~~~
(16) (a) Whangbo, M.-H. J . Chem. Phys. 1980, 73, 3854. (b) Brandow, B. H. Adu. Phys. 1977, 26, 651, and references therein. (17) {a) For a review of theoretical and experimental aspects of diradicals, see: "Diradicals"; Borden, W. T., Ed.; Wiley: New York, 1982. (b) Salem, L. "Electrons in Chemical Reactions: First Principles"; Wiley: New York, 1982; Chapter 3. (18) Borden, W. T.; Davidson, E. R. J . A m . Chem. SOC.1977, 99,4587.
(19) Some of the electron repulsion of the 'E' state is relieved by "symmetry breaking" when configuration interaction included in the description of wave function; see ref 17a. (20) Actually, this is only what is expected for the "zero-temperature" ground state. Whether a system will display ferromagnetism at finite temperatures depends on its dimensionality; e&, see: Landau, L. D.; Lifshitz, E. M."Statistical Physics"; Pergamon: Oxford, 1958.
J. Am. Chem. SOC.1985, 107, 6859-6865
where ( e d p ) = (ed + tp)/2 and Atdp = led - tpl. Eliminating (Ed!,) by choice of the energy zero, an exact solution for the density of states, g ( t ) , can be obtained from the formula for the band dispersion in eq A5:
K ( z ) is a complete elliptic integral of the first kind;21E is restricted to the range 0 IE2 I2 and is related to by the equation: €2
- (Atdp/2)2
E2 =
(A6b) 4@dp2
The T DOS has the form shown in Figure 2 where Aq, = 1.O eV, and = 1.8 eV. The form of the Wannier functions for the superdegenerate band can be determined from eq 10 and A l :
where the Wannier function considered is that centered on the origin (R = 0). After simplification, expressions for the Wannier function coefficients can be derived. The magnitude of the “inner” orbital coefficients illustrated in 15b was determined by numerical integration: (21) (a) Whittaker, E. T.; Watson, G. N . ”Modern Analysis”, 4th ed.; Cambridge University Press: Cambridge, 1927. (b) Gradshteyn, I. S.; Ryzhik, I. M. ‘Tables of Integrals, Series, and Products”, 4th ed.; Academic Press: New York. 1965.
6859
K ( x ) and E ( x ) are complete elliptic integrals of the first and second kind, respectively, and x 1 E (1 - x2)’j2. This value indicates that 89.3% of the electron density resides on the “inner” centers, which is to be compared to 100% implied by 15a. The remainder of the electron density resides on centers further removed. Appendix I1 All the computations were carried out using a program employing the extended Hiickel method22awhich may be used for both molecular and crystal calculations. It has been developed to its present state by M.-H. Whangbo, S. Wijeyesekera, M. Kertesz, C. N. Wilker, C. Zheng, and the author. The weighted Wolfsberg-Helmholz formula22b,cwas used for the Hij matrix elements. Pt and As parameters were taken from ref 6; Mo parameters are from ref 23. Exponents for P and As are originally from ref 24; the P p orbital energy was extrapolated from Herma~~-Skillman~~ tables so that is in alignment with EH parameters of adjacent elements of the periodic table. Exponents for the transition metals are originally from ref 26. See Table I . In the results reported for i[Pt3(As2)22-]and t [ M o H 2 P ]layers in Figures 4 and 5, matrix elements were computed between atoms separated by less than 6.55 A. Both are two dimensional hexagonal systems in which 1/12thof the Brillouin zone was covered by k-point meshes: 105 points were used for t[Pt3(As2)22-];300 points were used for i[MoH2P]. (22) (a) Hoffmann, R. J . Chem. Phys. 1963, 39, 1397. (b) Ammeter, J . H.; Biirgi, H.-B.; Thibeault, J. C.; Hoffmann, R. J . Am. Chem. SOC.1978, 100, 3686. (c) Summerville, R. H.; Hoffmann, R. Zbid. 1976, 98, 7240. (23) Kubicek, P.; Hoffmann, R.; Havlas, Z. Organometallics 1982, I , 180. (24) Clementi, E.; Roetti, C. AI. Nucl. Data Tables 1974, 1 4 , 177. (25) Herman, F.; Skillman, S. “Atomic Structure Calculations”; Prentice Hall: Englewood Cliffs, N.J., 1963. (26) Baranovskii, V. I.; Nikolskii, A. B. Teor. Eksp. Khim. 1967, 3, 527.
Gas-Phase Isotope Fractionation Factor for Proton-Bound Dimers of Methoxide Anions David A. Weilt and David A. Dixon*’ Contribution from the Department of Chemistry, University of California-Riverside, Riverside, California 92521, and E . I . du Pont de Nemours & Co., Central Research and Development Department, Experimental Station, Wilmington, Delaware 19898. Received June 8, 1984
Abstract: The gas-phase isotope fractionation factor, GgP, for A2L- (where A = M e 0 and L = H or D) has been measured by ion cyclotron resonance spectroscopy. The value for GBp is 0.33 f 0.06. The gas-phase value is compared with the solution measurements. Ab initio calculations on the electronic structure of the dimer have been performed at the DZ+P level. The dimer is found to be asymmetric with a central barrier to proton transfer on the order of 2 kcal/mol. The harmonic force field for the dimer has been determined with the 4-31G basis set. These results are used to calculate a theoretical value for GBP of 0.37 in excellent agreement with the experimental value. The calculated hydrogen bond strength is 25.6 kcal/mol.
Over the past 15 years, determinations of proton transfer equilibrium constants have increased the understanding of gasphase reaction dynamics and the effect that solvent molecules have on analogous solution-phase equilibrium.] In contrast, the rates of proton transfer reactions are not as easily determined in either the gas or solution phase, especially when the proton donor-acUniversity of California-Riverside. * E . I. du Pont de Nemours & Co. Contribution No. 3755.
0002-7863/85/1507-6859$01.50/0
ceptor separation is smalL2 A description of the potential energy surface for small proton-acceptor separation would aid in the (1) (a) D.H. Aue and M. T. Bowers In “Gas Phase Ion Chemistry”, M. T. Bowers, Ed., Academic Press, New York, 1979, Vol. 2, C9. (b) R. W. Taft In “Proton-Transfer Reactions”, E. Caldin and V. Gold, Eds., Wiley, New York, 1975, p 31. (c) E. M. Arnett, Ibid., p 79. (d) E. M. Arnett, Acc. Chem. Res., 6, 404 (1973). (e) R. W. Taft In “Progress in Physical Organic Chemistry”, R. W. Taft, Ed., Vol. 14, p 247, 1983, Wiley-Interscience, New York.
0 1985 American Chemical Society
